In this work, we are interested in studying the effects of \(\mathrm {\omega _{DE}}\)  and \(\bar{Q}\)  on the total matter power spectrum and on the CMB
temperature power spectrum. For this reason, we consider only the adiabatic perturbations, assume that \({T_{A}}^{\mu \nu }\)  is free of anisotropic
stress, and the arguments above discussed, we find the evolution equations for the density contrast perturbation
\(\delta _{A}\)  and the velocity perturbations \(\theta _{A}\)  in the IDE model from the general case presented in [1]}, [2]}
when \(Q^{\mu }_{A}\parallel U^{\mu }_{DM}\) ,
\(\frac{\mathrm {d}{\delta }_{DE}}{\mathrm {d}z}&=&-(\mathrm {R}{\bar{\rm I}}_{\rm Q}-3+3\mathrm {\omega _{DE}})\frac{\delta _{DE}}{(1+\mathrm {z})}+(1+\mathrm {\omega _{DE}})\frac{\theta _{DE}}{\bar{H}}\nonumber \\&&+\frac{\bar{H}\theta _{DE}}{(1+\mathrm {z})^{2}\mathrm {k^{2}}}\biggl [9(1-\mathrm {{\omega }_{DE}^{2}})-3(1+\mathrm {z})\frac{\mathrm {d}{\omega }_{DE}}{\mathrm {d}z}\nonumber \\&&+3\mathrm {R}{\bar{\rm I}}_{\rm Q}(1-\mathrm {\omega _{DE}})\biggr ]+3(1+\mathrm {\omega _{DE}})\frac{\mathrm {d}{\psi }}{\mathrm {d}z}+\nonumber \\&&+\frac{\mathrm {R}{\bar{\rm I}}_{\rm Q}\phi }{(1+\mathrm {z})}+\frac{\mathrm {R}{\bar{\rm I}}_{\rm Q}}{(1+\mathrm {z})}\delta _{DM},\\\frac{\mathrm {d}{\delta }_{DM}}{\mathrm {d}z}&=&\frac{\theta _{DM}}{\bar{H}}-\frac{{\bar{\rm I}}_{\rm Q}\phi }{(1+\mathrm {z})}+3\frac{\mathrm {d}{\psi }}{\mathrm {d}z}\,,\\\frac{\mathrm {d}{\delta }_{b}}{\mathrm {d}z}&=&\frac{\theta _{b}}{\bar{H}}+3\frac{\mathrm {d}{\psi }}{\mathrm {d}z},\\\frac{\mathrm {d}{\theta }_{DE}}{\mathrm {d}z}&=&-\frac{2{{\theta }_{DE}}}{(1+\mathrm {z})}\left(1+\frac{\mathrm {R}{\bar{\rm I}}_{\rm Q}}{(1+\mathrm {\omega _{DE}})}\right)-\frac{\mathrm {k^{2}}{\delta }_{DE}}{\bar{H}(1+\mathrm {\omega _{DE}})}\nonumber \\&&-\frac{\mathrm {k^{2}}\phi }{\bar{H}}+\frac{\mathrm {R}{\bar{\rm I}}_{\rm Q}{\theta }_{DM}}{(1+\mathrm {z})(1+\mathrm {\omega _{DE}})},\\\frac{\mathrm {d}{\theta }_{DM}}{\mathrm {d}z}&=&\frac{\theta _{DM}}{(1+\mathrm {z})}-\frac{k^{2}\phi }{\bar{H}},\\\frac{\mathrm {d}{\theta }_{b}}{\mathrm {d}z}&=&\frac{\theta _{b}}{(1+\mathrm {z})}-\frac{k^{2}\phi }{\bar{H}}.\) 
